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Convergence of the gradient projection method for generalized convex minimization. (English) Zbl 0963.90058
Summary: This paper develops convergence theory of the gradient projection method by {\it P. H. Calamai} and {\it J. J. Moré} [Math. Program. 39, 93-116 (1987; Zbl 0634.90064)] which, for minimizing a continuously differentiable optimization problem $\min \{f(x): x\in \Omega\}$ where $\Omega$ is a nonempty closed convex set, generates a sequence $x_{k+1}= P(x_k- \alpha_k \nabla f(x_k))$ where the stepsize $\alpha_k> 0$ is chosen suitably. It is shown that, when $f(x)$ is a pseudo-convex (quasi-convex) function, this method has strong convergence results: either $x_k\to x^*$ and $x^*$ is a minimizer (stationary point); or $\|x_k\|\to \infty$, $\arg\min \{f(x): x\in \Omega\}= \emptyset$, and $f(x_k) \downarrow \inf\{f(x): x\in \Omega\}$.

90C30Nonlinear programming
90C52Methods of reduced gradient type
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